How to Construct a Diamond or Other Gemstone Material in Cycles
You need dispersion
Dispersion is the phenomenon of multicolored highlights you see in most transparent substances, resulting from different colors (light frequencies) refracting at different angles (the prism effect).
Your reference image does not have this, which makes it immediately noticeable as CG. Instead I would suggest by starting with a reference like this one.
Kent Trammell explained this elegantly in a tutorial that was published on CGCookie but has since been removed. The first person to do it was probably Kirado, who shared the node setup.
Here is the basic node setup - note how the IOR values differ:

Taking it further...
After doing some renders with the above setup I felt that the material still looked too dark, so I decided to try mixing it with a plain white Glass Shader in a way that would not interfere with the colorful highlights. The &quot;Is Camera Ray&quot; output of the Light Path Node seemed to work pretty well for this. What it does is takes any area where the RGB summed color is not already reaching the camera, and allows the white (clear) Glass color to shine through just in those gaps.
The IOR of diamond is 2.4175 and the dispersion is .044. So by using these values, as well as a realistic diamond size, we can get something much more accurate. Make sure to apply the scale (CtrlA) of your diamond object if you want the refraction to be as accurate as possible.

Keep in mind that we are still &quot;faking it&quot; by having only three set IOR values (for red/green/blue). But practically speaking this turns out to be totally sufficient.
Geometry matters
With gems like diamonds, the cut determines the way the light gets reflected. Geometry plays an important role in the appearance of this material.
A page on diamond.com has diagrams that explain this really well:

Luckily, creating an ideal diamond mesh can be done in Blender in an instant. Just enable the Extra Objects (Mesh) add-on, and it will be an option in your Add Mesh menu.

Now pressing ShiftA in the Viewport gives you new options:

In my example I slightly beveled my diamond, which allowed me to use Smooth Shading instead of Flat Shading. This did away with the angular &quot;CG look&quot;, and scattered the light in more directions, brightening the gem. However, it has since been brought to my attention by user Thom Blair III that diamonds are sharp down to the molecular level, so a bevel is not accurate. Consider using Simple Subdivision instead.
Lighting matters
An HDRI environment map will provide a more diverse color palette of light for your scene, so consider using one if you want more realistic lighting.
If the environment color is not quite to your liking, either change to a better-suited environment map or adjust the color using a Hue/Saturation/Value Node. In my case the sky was too blue, so I reduced the saturation.

Rendering
A material like this requires a high sample count for it to look clear (I used 7500 samples!). *Update: With the new Denoise feature you don't need anywhere near this many samples now! Many thanks to Blender's developers.
If the gemstone will be part of a larger scene where other elements do not require such a high sample count, you can keep the gemstone on its own Render Layer to keep render times from becoming ridiculous. (Okay, I mean keep them less ridiculous.)

Setting Clamp Indirect to something like 10 also helps to reduce fireflies.

Compositing
If you want to adjust the exact color of your gem, or add bloom and streak effects, you can use the compositor to put on the final touches.
An example node setup for compositing:

Before compositing:

After compositing (final result):